Description

IMPETUS failure criterion is similar to the Cockcroft-Latham failure criterion. However, it has been equipped with one extra parameter $n$ that allows
for an anisotropic damage growth. The model is based on the assumption that defects deform with the material. It is further assumed that compressed
defects exposed to tensile loading are more harmful than elongated defects. The damage growth is assumed proportional to the maximum eigenvalue
$\hat\sigma_1$ of a distorted stress tensor $\hat{\mathbf\sigma}$.

where $\mathbf{\sigma}$ is the current stress tensor and $\mathbf{A}$ is a symmetric tensor describing the defect compression.
$\mathbf{A}$ is a function of the principal stretches $(\lambda_1, \lambda_2, \lambda_3)$ and of their corresponding eigenvectors
$(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)$.

Note that $\lambda_1$ is the maximum principal stretch. The formulation ensures that the damage growth is equivialent to
Cockcroft-Latham in proportional loading where $\lambda_1$ coincides with $\sigma_1$.

The optional irregularization parameters $(\alpha_{irr}, \beta_{irr})$ are used to amplify the damage growth in regions where the
Finite Element mesh is too coarse to accurately resolve the local variations of the strain field. The purpose is to significantly
reduce the mesh dependency.
Note that this irregularization procedure
currently only is implemented for 64-node cubic hexahedra. It has no effect on other element types. The amplified rate of damage growth
$\dot D_{amp}$ is defined as:

where $(\mathbf{\varepsilon}_a, \mathbf{\varepsilon}_b)$ are the average strain tensors at the eight element center IP's and eight corner IP's, respectively.
Hence, $\|\mathbf{\varepsilon}_a - \mathbf{\varepsilon}_b \|$ is a measure of the curvature of the strain field (in parametric space).